Craig Westerland

Abstract

We prove a higher chromatic analogue of Snaith’s theorem which identifies the
K–theory
spectrum as the localisation of the suspension spectrum of
ℂℙ∞ away
from the Bott class; in this result, higher Eilenberg–MacLane spaces play the role of
ℂℙ∞=K(ℤ,2). Using
this, we obtain a partial computation of the part of the Picard-graded homotopy of the
K(n)–local
sphere indexed by powers of a spectrum which for large primes is a shift
of the Gross–Hopkins dual of the sphere. Our main technical tool is a
K(n)–local
notion generalising complex orientation to higher Eilenberg–MacLane spaces. As for
complex-oriented theories, such an orientation produces a one-dimensional formal
group law as an invariant of the cohomology theory. As an application, we prove a
theorem that gives evidence for the chromatic redshift conjecture.